TASK 1: INTEGRATION Another application of integration is in probability. This task applies your algebra and calculus skills to this domain. If X is a continuous random variable with range [a, b] then the probability density function satisfies the following two properties: Property 1: f(x) ≥ 0 for all x = [a, b]. Property 2: * f(a)da = 1. a Question 1 (1) Explain why the following functions are, or are not, probability density functions: (a) f(x) = ² for x = [1, √e] (b) f(x) = 5 (9x²-x4) for x = [0,3] 1
TASK 1: INTEGRATION Another application of integration is in probability. This task applies your algebra and calculus skills to this domain. If X is a continuous random variable with range [a, b] then the probability density function satisfies the following two properties: Property 1: f(x) ≥ 0 for all x = [a, b]. Property 2: * f(a)da = 1. a Question 1 (1) Explain why the following functions are, or are not, probability density functions: (a) f(x) = ² for x = [1, √e] (b) f(x) = 5 (9x²-x4) for x = [0,3] 1
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![TASK 1: INTEGRATION
Another application of integration is in probability. This task applies your algebra
and calculus skills to this domain.
If X is a continuous random variable with range [a, b] then the probability density
function satisfies the following two properties:
Property 1: f(x) ≥ 0 for all x € [a, b].
Property 2: [² f
a
f(x) dx = 1.
Question 1
(1) Explain why the following functions are, or are not, probability density functions:
(a) f(x) = 2 for x = [1, √e]
(b) f(x) = 5 (9x² - x¹) for x = [0, 3]
1](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6954de0d-2b65-4359-813b-d12332619368%2F2bb68ef6-e6fc-40e4-a175-91807c6e1d95%2Fn2b2zls_processed.png&w=3840&q=75)
Transcribed Image Text:TASK 1: INTEGRATION
Another application of integration is in probability. This task applies your algebra
and calculus skills to this domain.
If X is a continuous random variable with range [a, b] then the probability density
function satisfies the following two properties:
Property 1: f(x) ≥ 0 for all x € [a, b].
Property 2: [² f
a
f(x) dx = 1.
Question 1
(1) Explain why the following functions are, or are not, probability density functions:
(a) f(x) = 2 for x = [1, √e]
(b) f(x) = 5 (9x² - x¹) for x = [0, 3]
1
![Given a probability density function f(x) with range [a, b], we can obtain the
probability that X is at most k
from the area of a region under the function.
k
P(X ≤ k) = f*
=fª*
a
To find the probability that c≤ X ≤ k:
f(x) dx
k
P(c ≤ x ≤ k) = [ * f(x)dx
C
Note: P(X k) = P(X ≤ k).
Question 2
• Find the probability P(x ≤ 3) given the probability density function you identi-
fied in Question 1.
Give an exact answer, and an approximate answer to two decimal places.
• Why is Property 1 in Question (1) essential for a function to be a probability
density function?
TASK 2:
In Weeks 7 and 8, we looked at the chain, product and quotient rules for finding the
derivative.
There are similar rules that can be used for integration. Investigate integration by
substitution and explain how you can use it to integrate. You can use this example,
[xe²
What differentiation rule, does the substitution rule remind you of?
re dr.
TASK 3:
(1) Sketch the graphs y = x²(x+2) and y = x(x+2) for the domain [-2, 1] shading
the areas between the curves.
Give all working for finding the intercepts and for finding and classifying the
stationary points.
Give all working for finding where the two graphs intersect.
(2) Find the total area of the regions between the curves in this domain.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6954de0d-2b65-4359-813b-d12332619368%2F2bb68ef6-e6fc-40e4-a175-91807c6e1d95%2Fsfqpcm_processed.png&w=3840&q=75)
Transcribed Image Text:Given a probability density function f(x) with range [a, b], we can obtain the
probability that X is at most k
from the area of a region under the function.
k
P(X ≤ k) = f*
=fª*
a
To find the probability that c≤ X ≤ k:
f(x) dx
k
P(c ≤ x ≤ k) = [ * f(x)dx
C
Note: P(X k) = P(X ≤ k).
Question 2
• Find the probability P(x ≤ 3) given the probability density function you identi-
fied in Question 1.
Give an exact answer, and an approximate answer to two decimal places.
• Why is Property 1 in Question (1) essential for a function to be a probability
density function?
TASK 2:
In Weeks 7 and 8, we looked at the chain, product and quotient rules for finding the
derivative.
There are similar rules that can be used for integration. Investigate integration by
substitution and explain how you can use it to integrate. You can use this example,
[xe²
What differentiation rule, does the substitution rule remind you of?
re dr.
TASK 3:
(1) Sketch the graphs y = x²(x+2) and y = x(x+2) for the domain [-2, 1] shading
the areas between the curves.
Give all working for finding the intercepts and for finding and classifying the
stationary points.
Give all working for finding where the two graphs intersect.
(2) Find the total area of the regions between the curves in this domain.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 2 images
Follow-up Questions
Read through expert solutions to related follow-up questions below.
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
